A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010). These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.Comment: 58 pages, 2 figure

Floquet topological transitions in extended Kane-Mele models with disorder

In this thesis we use Floquet theory to theoretically study the influence of circularly polarized light on disordered two-dimensional models exhibiting topological transitions. We find circularly polarized light can induce a topological transition in extended Kane-Mele models that include additional hopping terms and on-site disorder. The topological transitions are understood from the Floquet-Bloch band structure of the clean system at high symmetry points in the first Brillouin zone. The light modifies the equilibrium band structure of the clean system in such a way that the smallest gap in the Brillouin zone can be shifted from the M points to the K(Kʹ) points, the Γ point, or even other lower symmetry points. The movement of the minimal gap point through the Brillouin zone as a function of laser parameters is explained in the high frequency regime through the Magnus expansion. In the disordered model, we compute the Bott index to reveal topological phases and transitions. The disorder can induce transitions from topologically non-trivial states to trivial states or vice versa, both examples of Floquet topological Anderson transitions. As a result of the movement of the minimal gap point through the Brillouin zone as a function of laser parameters, the nature of the topological phases and transitions is laser-parameter dependent--a contrasting behavior to the Kane-Mele model.Physic

Using Infra-Red Beacons as Unobtrusive Markers for Mobile Augmented Reality

The main two approaches for vision based mobile augmented reality systems are either those employing fiducial markers or those which track natural features in the environment to estimate camera pose information. Whilst marker based systems are relatively simple to implement and are robust they present difficulties for wide scale deployment as they are obtrusive and their size is proportional to the distance from which they need to be used. However, the alternate approaches of marker less systems present significant computational challenges, can be highly problematic in poor light conditions, and are independent of scale. In the paper we present a novel solution using Infra Red LED’s as markers that overcomes many of these limitations in that they are: invisible to the human sight but can tracked by phone camera optics; can be used in varied light conditions; structured to provide scale; and significantly reduce the computational overhead

Maximal randomness expansion from steering inequality violations using qudits

We consider the generation of randomness based upon the observed violation of an Einstein-Podolsky-Rosen (EPR) steering inequality, known as one-sided device-independent randomness expansion. We show that in the simplest scenario -- involving only two parties applying two measurements with $d$ outcomes each -- that there exist EPR steering inequalities whose maximal violation certifies the maximal amount of randomness, equal to log(d) bits. We further show that all pure partially entangled full-Schmidt-rank states in all dimensions can achieve maximal violation of these inequalities, and thus lead to maximal randomness expansion in the one-sided device-independent setting. More generally, the amount of randomness that can be certified is given by a semidefinite program, which we use to study the behaviour for non-maximal violations of the inequalities.Comment: 6 pages, 1 figur

Loss-tolerant EPR steering for arbitrary dimensional states: joint measurability and unbounded violations under losses

We show how to construct loss-tolerant linear steering inequalities using a generic set of von Neumann measurements that are violated by $d$-dimensional states, and that rely only upon a simple property of the set of measurements used (the maximal overlap between measurement directions). Using these inequalities we show that the critical detection efficiency above which $n$ von Neumann measurements can demonstrate steering is $1/n$. We show furthermore that using our construction and high dimensional states allows for steering demonstrations which are also highly robust to depolarising noise and produce unbounded violations in the presence of loss. Finally, our results provide an explicit means to certify the non-joint measurability of any set of inefficient von Neuman measurements.Comment: 4+3 pages. v2: title changed. Results on unbounded violation of steering inequalities added. Accepted by PR

On the collision of two shock waves in AdS5

We consider two ultrarelativistic shock waves propagating and colliding in five-dimensional Anti-de-Sitter spacetime. By transforming to Rosen coordinates, we are able to find the form of the metric shortly after the collision. Using holographic renormalization, we calculate the energy-momentum tensor on the boundary of AdS space for early times after the collision. Via the gauge-gravity duality, this gives some insights on bulk dynamics of systems created by high energy scattering in strongly coupled gauge theories. We find that Bjorken boost-invariance is explicitely violated at early times and we obtain an estimate for the thermalization time in this simple system.Comment: 15 pages, 1 figure; v2: clarifications on boost-invariance and appendix added; v3: minor modifications, references added, matches published versio